This is often called "path method". In particular one has classically the Moser path method.

Let us prove for example (after Moser) that given a closed oriented $n$-manifold $M$ and two volume forms $\omega_0$, $\omega_1$ on $M$ with the same integral, there exists a diffeomorphism $f$ of $M$ such that $$f^*(\omega_1)=\omega_0$$ To this end, the path method considers $$\omega_t=(1-t)\omega_0+t\omega_1$$ and looks for a time-dependant vector field $X_t$ whose flow $\phi_t$ satisfies the condition $$\phi_t^*(\omega_t)=\omega_0$$ (Then, for $t=1$, $f=\phi^1$ will work)

The condition holds trivially for $t=0$. Deriving this condition with respect to $t$, one finds $$\phi_t^*(L_{X_t}\omega_t+\omega_1-\omega_0)=0$$ where $L$ is the Lie derivative. This amounts to $$L_{X_t}\omega_t=\omega_0-\omega_1$$ i.e. by Cartan's formula $$d\iota_{X_t}\omega_t=\omega_0-\omega_1$$ Since $\omega_0$ and $\omega_1$ have the same integral, they are cohomologous: one has a $(n-1)$-form $\alpha$ on $M$ such that $$\omega_0-\omega_1=d\alpha$$ Hence it is enough that $$\iota_{X_t}\omega_t=\alpha$$ The end of the argument is purely (multi)linear algebra: this equation admits for every time $t$ and at every point $x$ a unique solution $X_t(x)$ since $\omega_t(x)$ is a nonzero $n$-form on $T_xM$. An analogous method applies to symplectic forms, closed $1$-forms and contact forms. The path method also proves the $MJ^2$ Lemma and in particular the Morse Lemma.

This is often called "path method". In particular one has classically the Moser path method.

Let us prove for example (after Moser) that given a closed oriented $n$-manifold $M$ and two volume forms $\omega_0$, $\omega_1$ on $M$ with the same integral, there exists a diffeomorphism $f$ of $M$ such that $$f^*(\omega_1)=\omega_0$$ To this end, the path method considers $$\omega_t=(1-t)\omega_0+t\omega_1$$ and looks for a time-dependant vector field $X_t$ whose flow $\phi_t$ satisfies the condition $$\phi_t^*(\omega_t)=\omega_0$$ (Then, for $t=1$, $f=\phi^1$ will work)

The condition holds trivially for $t=0$. Deriving this condition with respect to $t$, one finds $$\phi_t^*(L_{X_t}\omega_t+\omega_1-\omega_0)=0$$ where $L$ is the Lie derivative. This amounts to $$L_{X_t}\omega_t=\omega_0-\omega_1$$ i.e. by Cartan's formula $$d\iota_{X_t}\omega_t=\omega_0-\omega_1$$ Since $\omega_0$ and $\omega_1$ have the same integral, they are cohomologous: one has a $(n-1)$-form $\alpha$ on $M$ such that $$\omega_0-\omega_1=d\alpha$$ Hence it is enough that $$\iota_{X_t}\omega_t=\alpha$$ The end of the argument is purely (multi)linear algebra: this last equation admits for every time $t$ and at every point $x$ a unique solution $X_t(x)$ since $\omega_t(x)$ is a nonzero $n$-form on $T_xM$. An analogous method applies to symplectic forms, nonsingular closed $1$-forms and contact forms. The path method also proves the $MJ^2$ Lemma (Lemma 3.2 in F. Laudenbach, A proof of Reidemeister-Singer's theorem by Cerf's methods. Ann. Fac. Sci. Toulouse Math. (6) 23 (2014), 1, 197–221, Arxiv 1202.1130); and in particular the Morse Lemma.